| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 8-s + 3·9-s + 10-s + 2·12-s + 4·13-s − 2·15-s + 16-s − 3·18-s − 20-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 2·30-s + 16·31-s − 32-s + 3·36-s + 4·37-s + 8·39-s + 40-s − 12·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 9-s + 0.316·10-s + 0.577·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s − 0.707·18-s − 0.223·20-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.365·30-s + 2.87·31-s − 0.176·32-s + 1/2·36-s + 0.657·37-s + 1.28·39-s + 0.158·40-s − 1.87·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.776497565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.776497565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.305587122869086271308905422277, −8.760711413234799859506496164896, −8.286230572976717874608236616986, −7.941231322257043910223245152113, −7.80207005021400136120224531772, −6.87333704174627289182797723156, −6.42217617652666865799421983967, −6.23991034800022982622767523992, −5.03421875022587082259817541459, −4.69263356392122883726033030816, −3.79769619454667096438258605845, −3.36585804145949552210873743484, −2.77580512142874145203019878589, −1.92677095659654979130865508090, −1.02898483249670331178782515928,
1.02898483249670331178782515928, 1.92677095659654979130865508090, 2.77580512142874145203019878589, 3.36585804145949552210873743484, 3.79769619454667096438258605845, 4.69263356392122883726033030816, 5.03421875022587082259817541459, 6.23991034800022982622767523992, 6.42217617652666865799421983967, 6.87333704174627289182797723156, 7.80207005021400136120224531772, 7.941231322257043910223245152113, 8.286230572976717874608236616986, 8.760711413234799859506496164896, 9.305587122869086271308905422277
